![]() The many exciting pipelines where such representations may be used (such as self-supervised learning, generative modelling, or reinforcement learning) are not our central focus \marginnoteThe same applies for techniques used for optimising or regularising our architectures, such as Adam ( kingma2014adam), dropout ( srivastava2014dropout) or batch normalisation ( ioffe2015batch). Length of a curve γ, discrete metric on edge ( u, v )īefore proceeding, it is worth noting that our work concerns representation learning architectures and exploiting the symmetries of data therein. Isomorphism between two different domainsĪctivation function (point-wise non-linearity) Scalar representing the jth component of a discrete domain signal X on element u ∈ Ωįunction on discrete domain signals that returns another discrete domain signal, as a matrix Vector representing a discrete domain signal X on element u ∈ Ω ![]() Matrix representing a signal on a discrete domain Signal on the domain of the form x : Ω → Cįunctions on signals on the domain of the form f : X ( Ω ) → Y The relation between these geometries is immediately apparent when considering the respective groups, because the Euclidean group is a subgroup of the affine group, which in turn is a subgroup of the group of projective transformations. This approach created clarity by showing that various geometries known at the time could be defined by an appropriate choice of symmetry transformations, formalized using the language of group theory.įor instance, Euclidean geometry is concerned with lengths and angles, because these properties are preserved by the group of Euclidean transformations (rotations and translations), while affine geometry studies parallelism, which is preserved by the group of affine transformations. properties unchanged under some class of transformations, called the symmetries of the geometry. In a research prospectus, which entered the annals of mathematics as the Erlangen Programme, Klein proposed approaching geometry as the study of invariants, i.e. ![]() Mathematicians and philosophers debating the validity of and relations between these geometries as well as the nature of the “one true geometry”.Ī way out of this pickle was shown by a young mathematician Felix Klein, appointed in 1872 as professor in the small Bavarian University of Erlangen. Towards the end of that century, these studies had diverged into disparate fields, with ‘geometry’ has been synonymous with Euclidean geometry, as no other types of geometry existed.Įuclid’s monopoly came to an end in the nineteenth century, with examples of non-Euclidean geometries constructed by Lobachevesky, Bolyai, Gauss, and Riemann. What is now called the ‘Erlangen Programme’ was actually a research prospectus brochure Vergleichende Betrachtungen über neuere geometrische Forschungen (“A comparative review of recent researches in geometry”) he prepared as part of his professor appointment. Klein indeed gave such a talk (though on December 7 of the same year), but it was for a non-mathematical audience and concerned primarily his ideas of mathematical education. Such a system would significantly accelerate the process of scientific discovery and engineering innovation.For nearly two millenia since Euclid’s Elements, the word \marginnoteĪccording to a popular belief, the Erlangen Programme was delivered in Klein’s inaugural address in October 1872. The goal is to create an AI system that can help scientists and engineers gain a deeper understanding of complex systems by automating the process of deriving equations and building test prototypes. However, this process can be labor-intensive and time-consuming, leading to the question of whether automation can assist in this endeavor.ĭumancic and his team want to explore the possibility of using artificial intelligence (AI) to help uncover explanations for the natural phenomena that surround us. Scientists and engineers follow this principle when they distil natural phenomena into concise mathematic equations and build test prototypes of complex machines. Scientists and engineers adopt this principle when they simplify natural phenomena and create mathematical equations to explain them. This statement emphasizes the importance of gaining a comprehensive understanding of a concept by starting from scratch, using fundamental principles and building blocks. Richard Feynman famously said "What I cannot create, I do not understand".
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